Problem: Given $ m \angle RPS = 2x + 65$, $ m \angle QPR = 4x - 4$, and $ m \angle QPS = 103$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {4x - 4} + {2x + 65} = {103}$ Combine like terms: $ 6x + 61 = 103$ Subtract $61$ from both sides: $ 6x = 42$ Divide both sides by $6$ to find $x$ $ x = 7$ Substitute $7$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 2({7}) + 65$ Simplify: $ {m\angle RPS = 14 + 65}$ So ${m\angle RPS = 79}$.